Speed can be zero, that is, the change in spatial position per unit of duration can be zero. Can the change in temporal position per unit of distance be zero, too? Let’s see.

First, the denominator cannot be zero. We cannot simply invert a zero speed because that would lead to a zero denominator, which is disallowed mathematically. The denominator is non-zero no matter what the measured quantity is.

Second, the units in the denominator are the reference for what the numerator is measured against. It’s as if the units keep ticking away while the numerator is measured. Since time is often in the denominator, the seconds, minutes, hours, etc. seem to be ticking away no matter what the value of the numerator is.

Third, in this case the distance units are in the denominator. The context is that distance units are ticking away while the duration is measured.

Here’s an example of what this means. Suppose you’re on a train going at a steady speed. The click clack of the train reminds you that it’s making distance. In your mind the click clack measures the distance away from your departure and closer to your destination.

Suppose a train comes up beside yours and goes at the same speed. You aren’t moving relative to the other train. But in units of distance what is the change in time? Since your motion is synchronized, there is no relative change in temporal position between the two trains. The relative change in time is zero, while the distance ticks off, click clack click clack.

Yes, a change in time per unit of distance can be zero.

*Related*